Comfort & Mill-Homogeneous Spaces

Topology and its Applications 21 (1985) 297-308 North-Holland

297

ON THE PRODUCT OF HOMOGENEOUS SPACES*

W.W. COMFORT

Mathematics Department, Wesleyan University, Middletown, CT 06457, USA

Jan VAN MILL Vrije Unioersiteit, Amsterdam, The Netherlands

Received 31 December 1984

Revised 12 March 1985

Within the class of Tychonoff spaces, and within the class of topological groups, most of the

natural questions concerning ‘productive closure’ of the subclasses of countably compact and

pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There

is a countably compact Tychonoff space X such that X XX is not pseudocompact; (2) [ZFC]

The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC +

MA] There are countably compact topological groups G,,, G, such that Go x G, is not countably

compact.

In this paper we consider the question of ‘productive closure’ in the intermediate class of

homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result

of V.V. Uspenskii, is this: In ZFC there are pseudocompact, homogeneous spaces X0, X, such

that X,, XX, is not pseudocompact; if in addition MA is assumed, the spaces X, may be chosen

countably compact.

Our construction yields an unexpected corollary in a different direction: Every compact space

embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number

LI there is a countably compact, homogeneous space whose Souslin number exceeds (r.

AMS (MOS) Subj. Class.: Primary S4D35, 54BlO; Secondary 54C15

homogeneous space countably compact retract universal topological property extension property Souslin number

Martin’s Axiom

pseudocompact

cellular number

0. Introduction. References to the literature

0.1. In this paper we consider only Tychonoff spaces; that is, completely regular,

Hausdorff spaces. The word ‘space’ is to be interpreted in this way throughout.

We use the symbol w to denote both the first infinite cardinal and the countably

infinite discrete space.

The Tech-Stone compactification of a space X is denoted PX; and for a con-

tinuous function f from a space X to a space Y we denote by 7 the (unique)

continuous extension off mapping pX into j?Y.

* This paper derives from the authors’ collaboration at the Vrije Universiteit of Amsterdam in

November, 1984. The first-listed author is pleased to thank the Department of Mathematics and Computer

Science at the Vrije Universiteit for generous hospitality and support.

298 W W. Comfort, J. Van Mill / Homogeneous spaces

For p E P(w)\o and Y the set of permutations of w, we write

It is clear for p, q E p(w)\w

that either T(p) = T(q) or T(p) n T(q) = 0. It is well known (see for example [7,

Section 91 or [35, Section 31 that (T(p)1 = c for each p E p(w)\@ and ){ T(p): p E

P(4\4 = 2’.

0.2. The symbol MA used in the abstract refers to Martin’s Axiom. The reader

unfamiliar with MA and its uses in topology might refer to M.E. Rudin [38,39] or

to Weiss [50].

0.3. Here are some bibliographical citations concerning the results (l), (2) and (3)

in our abstract.

(1) An example, credited to Novak [36] and Terasaka [45], is described in detail

by Gillman and Jerison [19, (9.15)].

(2) This is due to Comfort and Ross [lo]. A simplified proof based on an argument

of De Vries [48] appears in [5; (6.4), (2.6)].

(3) Such groups Gi are constructed by Van Douwen [ll] inside the compact

‘Boolean group’ (-1, +l}‘.

0.4. The theorem of Uspenskii [46] on which our work depends reads as follows:

Theorem ([46]). Let Y be a space with 1 Y\ = a 2 w and lef

H( Y) = {fE Y”: If’({y})( = a for all y E Y}.

Then :

(a) H( Y) is homogeneous ; (b) H(Y) is homeomorphic to Y x H( Y); and

(c) H(Y) admits a retraction onto a homeomorph of Y.

(Uspenskii’s theorem is easily proved. For (a) let g, h E H( Y) and let rr be a

permutation of LY such that 7-r 0 g-‘({y}) = hP’({y}) for each y E Y; the mapf+fo rr

is a homeomorphism of H(Y) onto H(Y) such that g+ h. For (b), from which (c)

follows, it is enough to write as usual

(Yfl=(Yu{a}={~: 5dLy)

and to consider the map

Y x H( Y) + {g E Ya+‘: Ig-‘({y})l= (Y for all y E Y}

givenby (y,f)+g with g(O=f(5) for &<a, g(a)=y.)

Given an infinite space Y, we use throughout this paper the symbol H(Y) to

denote the homogeneous space defined in the statement of Theorem 0.4.

W. W. Comfort, J. Van Mill / Homogeneous spaces 299

1. On the product of pseudocompact, homogeneous spaces

Let us begin with a warm-up exercise.

1.1. Theorem. [ZFC] There arepseudocompact, homogeneous spaces X0, X1 such that

X0 XX, is not pseudocompact.

Proof. Choose pO, p1 E p(o)\0 such that T(p,) n T( pl) = 0, and for i = 0, 1 define

Y, = w u T( pi) and Xi = H( Y).

In Y0 X Y,, the family { Un: n < w} defined by U,, = {(n, n)} is an infinite, locally

finite family of non-empty open sets. Thus Y0 x Y, is not pseudocompact. (Another

way to see it: The function f: Y0 X Y1 + R defined by f((n, n)) = n, f- 0 elsewhere,

is continuous and unbounded.) Since Xi retracts onto (a homeomorph of) yi, the

space X0 x X1 retracts onto Y0 x Y,. Thus X0 XX, is not pseudocompact.

It remains to see that the spaces Xi are pseudocompact. The argument is in two

stages. (1) First, Yl is pseudocompact for all cardinals K. For a proof, based on

the observation [21] that a product of spaces is pseudocompact if and only if each

countable subproduct is pseudocompact, see Frolik [ 151. (2) Secondly, Xi is a subset

of YF with the property that TAXi = Yf for each countable A c c (that much is

obvious), and each such subspace of a pseudocompact product is itself pseudocom-

pact. (This latter assertion is easily proved. See for example [6,26] or [9, (3.3(b))]

for generalizations.) 0

1.2. Let us note in passing that an attempt to use simply the argument of Theorem

1.1 to produce in ZFC two countably compact, homogeneous spaces X,,, X1 such

that X0 XX, is not pseudocompact cannot succeed.

Theorem. If Y is an infinite space then H(Y) is not countably compact.

Proof. Indeed, H(Y) is sequentially dense in Y” (with (Y = ) YI) in the sense that

for every f E Y* there is a sequence {fn: n < w} in H( Y) such that fn + f in Y*. To

see this, let {A(y, n): y E Y, n < w} be a faithfully indexed partition of (Y into sets

of cardinality (Y, and given f E Y” define fn E Y” by

if rl E A(Y, n 1,

fn(77) = {;( 7) otherwise. q

Evidently we will have to extend Uspenskii’s construction before dealing success-

fully with countably compact spaces.

300 W. W. Comfort, J. Van Mill / Homogeneous spaces

2. Universal topological properties

Following Van der Slot [43], we say that a property 9 possessed by some spaces

is a universal topological property provided:

(a) every compact space has 9;

(b) the product of any set of spaces with 9 has 9; and

(c) every closed subspace of a space with 9 has 9’.

Within our context (Tychonoff spaces), the universal topological properties are

exactly the topological extension properties defined and studied by Woods [51].

The following theorem, a special case of the adjoint functor theorem of Freyd

[13,14], is accessible through arguments given by Kennison [31], Van der Slot

[43,44], Herrlich [23], Herrlich and Van der Slot [24], Franklin [ 121 and Woods

[51]; a systematic exposition is given in [8].

2.1. Theorem. Let 9 be a universal topological property, and for every space Y define

&D(Y)=n{z: YcZcpY,Zhas CP}.

Then

(4 PA Y) has p; (b) Y is dense in pP( Y) ;

(c) for every continuous f: Y+ X such that X has 9, the function f satisJies

f [PA VI = X ; and (d) up to a homeomorphism Jixing Ypointwise, &.( Y) is the only space satisfying

(4, (b) and (c).

The following simple result has been noted and used, at least for special instances

of 9, by many authors.

2.2. Lemma. Let 9 be a universal topological property and let h be a homeomorphism

of a space Y onto Y. Then h extends to a homeomorphism of&(Y) onto &(Y).

Proof. Let j = 61/3,(Y) and k =h-‘I&.( Y). By Theorem 2.1(c) both k 0 j and jo k

map &(Y) continuously into itself, and both are the identity on Y. Thus both k 0 j

and j 0 k are the identity on &(Y), so j is a homeomorphism of the kind

required. q

In the following lemma we associate with each P-space Y an enveloping P-space

I(Y) to which Y-homeomorphisms extend.

2.3. Lemma. Let 9 be a universal topological property. For every space Y with 5’ there

is a space I( Y), containing a homeomorph F of Y, such that:

(a) I(Y) has P’;

(b) Z(Y) admits a retraction onto Y;

(c) every homeomorphism of y onto F extends to a homeomorphism of I(Y) onto

I( Y) ; and


(d) for every p, q E y there is a homeomorphism h of I(Y) onto I(Y) such that

h(p) = q.

Proof. Set I(Y) = &+( Y x H( Y)). Fix f~ H(Y), and set Y= Y x(f).

Statement (a) is given by Theorem 2.1(a). According to Theorem 2.1(c) the

retraction r: Y x H( Y) + Y given by (y, g) + (y, J) satisfies f[ I( Y)] = Y; this proves

(b). We prove (c). Let h be a homeomorphism of Y onto Y and denote again by h

its natural extension (given by (y, g) + (h(y,f), g)) from Y x H( Y) onto Y x H( Y).

An appeal to Lemma 2.2 completes the proof.

For (d) use Theorem 0.4(a), (b) to find a homeomorphism h of Y x H( Y) onto

Y x H( Y) such that h(p) = q, and again apply Lemma 2.2. 0

2.4. We remark that for universal topological properties 9 and spaces Y, 2 the

relation pP(YXZ)=&(Y)Xp,(Z) can fail, even when Y has B. Indeed if 9 is

the most important of all such properties, compactness, and Y is an infinite compact

space, then according to Glicksberg’s theorem [21] an infinite space 2 satisfies

&( Y xZ) = Y x/?~(Z) if and only if Z is pseudocompact.

2.5. It would be nice to embed every P-space as a retract into a homogeneous

P-space, but our methods appear to be inadequate to this task. The argument we

give in Theorem 2.6 applies just when 9 is cardinally determined in the sense of

the following definition.

Definition. Let 9’ be a property possessed by some spaces and let LY be an infinite

cardinal. Then 9’ is a-determined if, for each space Y, the following condition is

satisfied: Y has 9 if and only if every closed subset F of Y such that IFI s (Y has

P. If 9 is a-determined for some infinite cardinal cx then ?? is cardinally determined.

We illustrate this notion by mentioning some properties which are cardinally

determined and some which are not. (We remind the reader that all spaces in this

paper are Tychonoff spaces. Thus for Y a space and AC Y we have always

Icl yA( s 2*‘?)

(a) Compactness and realcompactness, both universal topological properties, are

not cardinally determined. To see that neither property is a-determined when (Y 3 w,

let D be a discrete space with ID[> LY and define

Y = u {cl,& A c 0, IAl c a}.

If F is a closed subset of Y such that IFI < LY then for y E F there is A, c D such

that [A,,] G (Y and y E clpD Ay Writing A = lJvEFAy we have IAl s (Y and F c clpD A c

Y; thus F is compact (and hence realcompact). But Y itself is not realcompact.

To show this it is sufficient, according to Hewitt [25], to show that for some p E /3Y\ Y

no continuous f: /3 Y -+ [0, l] satisfies f(p) = 0 and f> 0 on Y (For proofs of this


and of other characterizations of realcompactness, the interested reader should

consult Gillman and Jerison [ 191 or Walker [49].) Indeed, for no p E /3 Y\ Y does

such a function f exist. For if f is given then since D is dense in Y for n < o there

is x, E D such that f(x,) < l/n, and then with B = {x,: n < OJ} we have that clpD B

is compact, f > 0 on clpD B since clPD B = Y, and infflclp,B] = 0.

(b) For an infinite cardinal y, a space Y is said to be y-bounded if cly A is

compact for each A c Y such that IAl 6 y. It is easy to see (and it has been noted

by several mathematicians) that y-boundedpess is a universal topological property

(Y 2 w). An argument much like that of the preceding paragraph shows for each

y> o that y-boundedness is cardinally determined (in fact, y-boundedness is

a-determined with cy = 2*‘).

(c) Countable compactness, though not a universal topological property, is 2’-

determined. In Section 3 for p E p(w) we will define p-compactness: it will be clear

that each of these is 2’-determined.

In the following construction, given a universal topological property 5P we associ-

ate with each space 2 with 9 the space I(Z) given by Lemma 2.3 ; in the interest

of notational simplicity we identify Z with its homeomorph Z c I(Z).

2.6. Let B be a universal topological property and let Y be a space with PP. For

use in the following theorem we define by transfinite recursion: a space X0 and a

function r, : X0 + Y; for each successor ordinal ,$+ 1, a space Xc+, and a function

r5+1. .X,+,+ Y; and for each limit ordinal 5, spaces X; and X, and functions

r;:X;* Y and re:X,+ Y.

We proceed as follows. Let X,, = Z( Y) and let r, be the retraction given by Lemma

2.3(b) ; for the successor ordinal [+ 1, X, and rC having been defined, let X,,, =

Z(X,), let s: X,,, + X, be the retraction given by Lemma 2.3(b), and let re+, = rC 0 s;

and for a limit ordinal 5, X, and ri having been defined for all 5 < 5, let Xl = lJ5<5 X,

topologized so that a subset U of X; is open if and only if U n X, is open in X,

for each 5 < 5, let r; = U t<5 ri, let X, = Z(X;) and, noting that r; is continuous from

Xk onto Y, let rC be the restriction to X, of the function 2: p(X;) + BY

Theorem. Let 9 be a universal topological property and let Y be a space with 9. Then

the spaces and functions X,, X;, re, r-i de$ned above satisfy:

(a) each X, has 9, and r( is a retraction of X, onto Y;

(b) each X; (5 a limit ordinal) is homogeneous, and r; is a retraction of Xi onto Y;

(c) if 9’ is a-determined (o 2 w) then X&+ has 9’.

Proof. For each 5 there is Z such that X, = p?(Z), so X, has P. The statements

about rC and r; are clear by induction, using (for limit 5) Theorem 2.1(c) and the

fact that Y has 9’.

To see that X; is homogeneous for limit 5, let p, q E X; and find n < 5 such that

p, q E X,. By Lemma 2.3(d) applied to X,, there is a homeomorphism h,,, of X,,,

W. W. Comfort, J. Van Mill / Homogeneous spnces 303

onto X,,, such that h,+l(p) = q. A simple inductive argument will yield a homeo-

morphism h of X; onto X; such that h,,, c h: Use Lemma 2.3(d) for non-limit

ordinals JJ+ 1 with q s J’< 5 to define a homeomorphism hi+, of X,,, onto X,,,

such that h, c h5+,, and for limit ordinals C with n < 5 G [ let h; = IJiCC hi : X; + X;

and define h,: X, + X, using Lemma 2.3(d); finally, take h = h,.

To verify (c) it is enough to show that if 9 is a-determined and F is a closed

subset of X&+ then F has 9. This is clear because there is [< (Y+ such that F c X,,

X, has ??, and F is closed in X,.

The proof is complete. 0

3. Homogeneous spaces with large Souslin number

Before passing in Section 4 to the construction in ZFC+MA of homogeneous,

countably compact spaces whose product is not pseudocompact, we pause briefly

to note some consequences of the foregoing theorem. It is convenient to have some

other universal topological properties at our disposal, as follows.

Definition. Let p E p(w). A space X is p-compact if for everyf: w + X the continuous

extension J: /?( w ) + PX satisfies f(p) E X.

The concept of p-compactness for p E p(w) was discovered, investigated and

exploited by Bernstein [l] in connection with problems in the theory of non-standard

analysis; it may be noted that the p-limit concept of Bernstein [l] coincides in

important special cases with the ‘producing’ relation introduced by Frolik [16, 171,

with one of the orderings considered by Katetov [29,30], and with a definition

given independently by Saks [41].

The following result is due to Bernstein [ 11. Proofs are available also in [20,42,47].

3.1. Lemma ([ 11). For p E p (w )\w, p-compactness is a universal topological property.

When 9 is p-compactness and Y is a space, we write p,(Y) in place of/&(Y).

3.2. Corollary. Let Y be a compact space. Then:

(a) Y embeds as a retract in a u-compact (hence, Lindeliif) homogeneous space;

(b) for every cardinally determined universal topological property 9, Y embeds as

a retract in a homogeneous space with 9 ; and

(c) for p E P(w)\w, Y embeds as a retract in a p-compact (hence, countably

compact), homogeneous space.

Proof. The compact space Y has every universal topological property. Statement

(a) then follows from Theorem 2.6; the required enveloping a-compact space is the


space X, of Theorem 2.6. Statement (b) follows similarly from Theorem 2.6, and

(c) is a special case of (b) since p-compactness is 2’-determined. 0

3.3. It has come to our attention that Corollary 3.2(a) has been proved already by

Okromeshko [37]. For a number of properties Q (e.g., the Lindelof property,

paracompactness and the like) he is able to embed every space with Q as a retract

of a homogeneous space with Q. His methods are quite different from ours, and so

far as we can determine the only point of overlap or duplication is Corollary 3.2(a)

above.

Definition. Let Y be a space. A cellularfamily in Y is a family of pairwise disjoint,

non-empty open subsets of Y.

The Souslin number of Y, denoted S(Y), is the cardinal number

S( Y) = min{ cy : every cellular family %! of Y has I%( < 0).

A closely related concept, the cellular number of Y, denoted c(Y), is the cardinal

number

c( Y) = sup{(~: some cellular family % of Y has 1%) = a}.

For references to the literature concerning these numbers and for proofs of many

theorems concerning them, see [7,9,27].

3.4. Corollary. For every cardinal number (Y there are homogeneous, u-compact spaces, and there are homogeneous, countably compact spaces, whose Souslin number exceeds

a.

Proof. The relation S(X) 3 S(Y) holds whenever Y is the image of X under a

continuous surjection. Thus the two statements follow from parts (a) and (c)

respectively of Corollary 3.2, upon taking for Y (for example) the space /3D with

D discrete, IDI = (Y. 0

4. On the product of countably compact, homogeneous spaces

The Rudin- Keisler ( pre-) order < on p(w)\@ is defined as follows: r <p if there

is f: w + w such that f(p) = r. In the proof of Theorem 4.1 we need the existence

of elements p in p(w)\0 which are <-minimal in the sense that if r E p(w)\w and

r < p then T(r) = T(p). (For a discussion of < and of several conditions equivalent

to <-minimality, see [7, (9.6)].) K unen [32] has shown that the existence of


<-minimal p E p(w)\w cannot be proved in ZFC, but it is known that such p exist

in any model of ZFC satisfying CH [3; 4; 7, (9.13)]] or MA [2,39].

4.1. Theorem. [ZFC+MA] There are countably compact, homogeneous spaces X,,

X, such that X, XX, is not pseudocompact.

Proof. From MA there are by [2,39] <-minimal po, p1 E p(w)\w such that T( po) n

T(p,) = 0. We set Y(i) = p,,(w) and, noting that p,-compactness is a 2’-determined

universal topological property, we use Theorem 2.6 to define spaces X(i) (of the

form X[,r,+) such that X(i) is pi-compact, X(i) is homogeneous, and X(i) retracts

onto Y(i) (i = 0, 1). Since X(0) xX( 1) then retracts onto Y(0) X Y(l), to complete

the proof it is enough to show that Y(0) x Y(1) is not pseudocompact.

For this as in Theorem 1.1 it is enough to show {(n, n): n < w} is closed in

Y(0) x Y(l), i.e., that Y(0) n Y(1) = o. We show in fact, writing

E(i) = w u IJ {cl pcwjA: A= T(Pi), IA] s ~1, (i = O, l),

that (a) E(i) is pi-compact (hence, Y(i)= E(i)) and (b) E(O)nE(l)=w.

(a) Letf:w+E(i). If {n<w:f(n)~~}~p~ thenf(pi)<pi and hence

from the =G -minimality of pti If {n<w:f(n)EW}$pi then with F=

{n < w:f(n) E E(i)\w} we have FEpi and for n E F there is countable A,, c T(p,)

such that f(n) E dpcW,A,; then with A = lJneF A, we have IAl s w and hence

f(~z) E cl,+,f[Fl= cl,qujA~ E(i),

as required.

(b) Let Ai c T( pI) with IAil G w, i = 0, 1. Each point of Ai is a P-point of p(w)\w

(see [7], for example) and hence Aon c~~(_,~A, = ~l~(~,A,n A, = 0. It then follows,

as noted in a related context by Frolik [17, 181 (see also [19, (14N.5); 49, (1.64);

7, (16.15); 5, (2.10)] that cl p(wJAo n cI~(~~A, = 0, as required. 0

5. Some questions

Here we list some questions which our arguments appear unable to settle.

5.1.(a) Is the conclusion of Theorem 4.1 available in ZFC? Are there in ZFC

countably compact, homogeneous spaces X0, X, such that X0 xX, is not countably

compact?


(b) Clearly the argument of Theorem 4.1 applies whenever one has two cardinally

determined universal topological properties LPO and 8,, each implying countable

compactness, for which there exist spaces X(i) with pi (i = 0,l) such that X(0) x

X(1) is not pseudocompact. Can one find such properties pi in ZFC?

We have referred already to Kunen’s result [32] that there is a model of ZFC in

which no p E p(w)\w is <-minimal. It is not known whether in ZFC one can find

po, p, E p(w)\w such that no qE p(w)\w satisfies q=Sp, (i=O, 1). (For a related

problem see Van Mill [35, (5.2)].)

(c) It may be difficult to respond positively to Questions 5.1(a) and 5.1(b) without

recourse to the theory of p(w)\w. Indeed, Kannan and Soundararajan [28] have

shown for a vast class of properties 9 closely related to the universal topological

properties-the class of so-called PCS properties-that a space has 9 if and only

if it is p-compact for each p in some set or class of ultrafilters (living not necessarily

on w but on various discrete spaces). This result has been extended and formulated

in a categorical context by Hager [22]. For other results on spaces required to be

p-compact simultaneously for various p, see Woods [51] and Saks [42].

5.2. It appears that neither our methods nor those of Okromeshko [37] are strong

enough to furnish a positive response to the following question, which was brought

to our attention some years ago by Eric Van Douwen: Is there for every cardinal

number LY a homogeneous, compact space X such that S(X) > (Y? The Haar measure

theorem shows S(G) s w+ for every compact topological group G; Maurice [33]

and Van Mill [34] have furnished in ZFC examples of compact, homogeneous

spaces X such that S(X) = c+. We do not even know whether in some or every

model of ZFC there is a compact, homogeneous space X such that S(X) > ct.

5.3. Is there a pseudocompact (or even: countably compact) homogeneous space

X such that X xX is not pseudocompact?

5.4. Given an a-determined universal topological property 9 and a space Y with

9 such that y = w( Y) (the weight of Y), we have in Theorem 2.6 embedded Y as

a retract into a homogeneous g-space X = X, ’ + such that 1X1= l,+(y) = a,+(1 Y]).

(Here as usual for each cardinal y 2 w the beth cardinals ItC(y) are defined by

3,,(y) = y, let,(y) = 2’~(~) for ordinals e> 0, and l*(y) =CT+ l,(y) for limit

ordinals 5. When y = w(Y) the relation 1,(y) = >,,,(I Yl), and hence l,+(y) =

1,+(( Y]), results from the inequalities ) YI s 2’, y s 21yi.) It seems a long way from

y to 1,+(y). Problem: Given such (9, (Y, Y, y), find the minimal cardinal 6 =

6(9, (Y, Y, y) for which Y embeds as a retract into a homogeneous L?‘-space X such

that IX]= 6.

5.5. Remark (added March, 1985). We have been told informally that S. Shelah

has announced in conversation the following result: It is consistent with ZFC that

(*) for all po, p1 E p(w)\w there is r E p(w)\w such that r=Spti


It is clear that in a model with (*) no spaces X(i) as in Theorem 4.1 can exist

of the form X(i) = X;,c,+ with Y(i) =&,(w), pi E p(w)\w.

References

[l] A.R. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1970) 185-193.

[2] D.D. Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970) l-24.

[3] G. Choquet, Construction d’ultrafiltres sur N. Bull. Sci. Math. (2) 92 (1968) 41-48.

[4] G. Choquet, Deux classes remarquables d’ultrafiltres sur N, Bull. Sci. Math. (2) 92 (1968) 143-153.

[5] W.W. Comfort, Topological groups, in: K. Kunen and J.E. Vaughan, eds., Handbook of Set-theoretic

Topology (North-Holland, Amsterdam, 1984) 1143-1263.

[6] W.W. Comfort and S. Negrepontis, Continuous functions on products with strong topologies, in:

J. Novak, ed., General Topology and its Relations to Modern Analysis and Algebra, II1 (Proc. 1971

Prague Topological Symposium) (Academia, Prague, 1972) 89-92.

[7] W.W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Grundlehren der math. Wissen-

schaften in Einzeldarstellungen, Vol. 211 (Springer, Berlin, 1974).

[E] W.W. Comfort and S. Negrepontis, Continuous Pseudometrics, Lecture Notes in Pure and Applied

Mathematics, Vol. 14 (Dekker, New York, 1975).

[9] W.W. Comfort and S. Negrepontis, Chain Conditions in Topology, Cambridge Tracts in Mathe-

matics, Vol. 79 (Cambridge University Press, London, 1982).

[lo] W.W. Comfort and K.A. Ross, Pseudocompactness and uniform continuity in topological groups,

Pacific J. Math. 16 (1966) 483-496.

[ll] E.K. Van Douwen, The product of two countably compact topological groups, Trans. Amer. Math.

Sot. 262 (1980) 417-427. [ 121 S.P. Franklin, On epi-reflective hulls, Gen. Topology Appl. 1 (1971) 29-31.

[13] P.J. Freyd, Functor theory, Doctoral Dissertation, Princeton University, 1960.

[14] P.J. Freyd, Abelian Categories: An Introduction to the Theory of Functors (Harper and Row, New

York, 1964).

[15] Z. Frolik, On two problems of W.W. Comfort, Comment. Math. Univ. Carolinae 8 (1967) 139-144.

[ 161 Z. Frolik, Types of ultralilters on countable sets, in: General Topology and its Relations to Modern

Analysis and Algebra, II (Proc. Second Prague Topological Symposium, 1966) (Academia, Prague,

1967) 142-143.

[17] Z. Frolik, Sums of ultrafilters, Bull. Amer. Math. Sot. 73 (1967) 87-91.

[ 181 Z. Frolik, Maps of extremally disconnected spaces, theory of types, and applications, in: General

Topology and its Relations to Modern Analysis and Algebra, III (Proc. 1968 Kanpur Conference) (Academia, Prague, 1971) 131-142.

[19] L. Gillman and M. Jerison, Rings of Continuous Functions, (Van Nostrand, New York, 1960).

[20] J. Ginsburg and V. Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975) 403-418.

[21] I. Glicksberg, Stone-Tech compactifications of products, Trans. Amer. Math. Sot. 90 (1959) 369-382.

[22] A.W. Hager, A description of HSP-like spaces, and applications, Pacific J. Math. (1985), to appear. [23] H. Herrlich, G-kompakte R&me, Math. Z. 96 (1967) 228-255.

[24] H. Herrlich and J. Van der Slot, Properties which are closely related to compactness, Proc. Nederl.

Akad. Wetensch. Series A 70 = Indag. Math. 29 (1967) 524-529. [25] E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Sot. 64 (1948) 45-99.

[26] M. HuSek, Products as reflections, Comment. Math. Univ. Carolinae 13 (1972) 783-800. [27] I. Juhasz, Cardinal Functions in Topology-Ten Years Later, Mathematical Centre Tracts 123

(Mathematisch Centrum, Amsterdam, 1980).

[28] V. Kannan and T. Soundararajan, Properties that are productive, closed-hereditary and surjective, Topology Appl. 12 (1981) 141-146.

[29] M. Katetov, Characters and types of point sets, Fund. Math. 50 (1961/62) 369-380 (in Russian). [30] M. Katetov, Products of filters, Comment. Math. Univ. Carolinae 9 (1968) 173-189.


[31] J.F. Kennison, Reflective functors in general topology and elsewhere, Trans. Amer. Math. Sot. 118 (1965) 303-315.

[32] K. Kunne, Ultrafilters and independent sets, Trans. Amer. Math. Sot. 172 (1972) 299-306. [33] M.A. Maurice, Compact Ordered Spaces, Mathematical Centre Tracts 6 (Mathematisch Centrum,

Amsterdam, 1964).

[34] J. Van Mill, A homogeneous Eberlein compact space which is not metrizable, Pacific J. Math 101

(1982) 141-146.

[35] J. Van Mill, An introduction to PO, in: K. Kunen and J.E. Vaughan, eds., Handbook of Set-theoretic

Topology (North-Holland, Amsterdam, 1984) 503-567. [36] J. Novak, A paradoxical theorem, Fund. Math. 37 (1950) 77-83.

[37] N.G. Okromeshko, On retractions of homogeneous spaces, Doklady Akad. Nauk SSSR 268 (1983)

548-550 (in Russian; English translation: Soviet Math. Doklady 27 (1983) 123-126).

[38] M.E. Rudin, Lectures on Set Theoretic Topology, Regional Conferences Series in Mathematics 23

(American Mathematical Society, Providence, RI, 1975).

[39] M.E. Rudin, Martin’s axiom, in: J. Barwise, ed., Handbook of Mathematical Logic, Studies in

Logic and the Foundations of Mathematics, Vol. 90 (North-Holland, Amsterdam, 1977) 491-501.

[40] W. Rudin, Homogeneity problems in the theory of Tech compactifications, Duke Math. J. 23 (1956)

409-419; 633. [41] V. Saks, Countably compact groups, Doctoral Dissertation. Wesleyan University, Middletown, CT,

1972.

[42] V. Saks, Ultrafilter invariants in topological spaces, Trans. Amer. Math. Sot. 241 (1978) 79-97.

[43] J. Van der Slot, Universal topological properties, ZW 1966-011, Math. Centrum, Amsterdam, 1966. [44] J. Van der Slot, Some properties related to compactness, Doctoral Dissertation, University of

Amsterdam, Math. Centrum, Amsterdam, 1968.

[45] H. Terasaka, On product of compact spaces, Osaka Math. J. 4 (1952) 11-15.

[46] V.V. Uspenskii, For any X, the product X x Y is homogeneous for some Y, Proc. Amer. Math.

Sot. 87 (1983) 187-188. [47] J.E. Vaughan, Countably compact and sequentially compact spaces, in: K. Kunen and J.E. Vaughan,

eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam, 1984) 569-602. [48] J. De Vries, Pseudocompactness and the Stone-tech compactification for topological groups, Nieuw

Archief voor Wiskunde (3) 23 (1975) 35-48.

[49] R.C. Walker, The Stone-tech Compactilication, Ergebnisse der Mathematik und ihrer Grenzgebiete,

Vol. 83 (Springer, Berlin, 1974).

[50] W.A.R. Weiss, Versions of Martin’s Axiom, in: K. Kunen and J.E. Vaughan, eds., Handbook of

Set-theoretic Topology (North-Holland, 1984) 827-886.

[51] R. Grant Woods, Topological extension properties, Trans. Amer. Math. Sot. 210 (1975) 365-385.

Comfort & Mill-Homogeneous Spaces

Documents

Transcript of Comfort & Mill-Homogeneous Spaces